Optimization and Control (math.OC)

Abstract: The current paper initially studies the optimal control of linear $\psi$-Hilfer fractional derivatives with state-dependent control constraints and optimal control for a particular type of cost functional. Then, we investigate the approximate controllability of the abstract fractional semilinear differential inclusion involving $\psi$-Hilfer fractional derivative in reflexive Banach spaces. It is known that the existence, uniqueness, optimal control, and approximate controllability of fractional differential equations or inclusions have been demonstrated for a similar type of fractional differential equations or inclusions with different fractional order derivative operators. Hence it has to research fractional differential equations with more general fractional operators which incorporate all the specific fractional derivative operators. This motivates us to consider the $\psi$-Hilfer fractional differential inclusion. We assume the compactness of the corresponding semigroup and the approximate controllability of the associated linear control system and define the control with the help of duality mapping. We observe that convexity is essential in determining the controllability property of semilinear differential inclusion. In the case of Hilbert spaces, there is no issue of convexity as the duality map becomes simply the identity map. In contrast to Hilbert spaces, if we consider reflexive Banach spaces, there is an issue of convexity due to the nonlinear nature of duality mapping. The novelty of this paper is that we overcome this convexity issue and establish our main result. Finally, we test our outcomes through an example.

Abstract: Mycobacterium leprae is a bacteria that causes the disease Leprosy (Hansen's disease), which is a neglected tropical disease. More than 200000 cases are being reported per year world wide. This disease leads to a chronic stage known as Lepra reaction that majorly causes nerve damage of peripheral nervous system leading to loss of organs. The early detection of this Lepra reaction through the level of bio-markers can prevent this reaction occurring and the further disabilities. Motivated by this, we frame a mathematical model considering the pathogenesis of leprosy and the chemical pathways involved in Lepra reactions. The model incorporates the dynamics of the susceptible schwann cells, infected schwann cells and the bacterial load and the concentration levels of the bio markers $IFN-\gamma$, $TNF-\alpha$, $IL-10$, $IL-12$, $IL-15$ and $IL-17$. We consider a nine compartment optimal control problem considering the drugs used in Multi Drug Therapy (MDT) as controls. We validate the model using 2D - heat plots. We study the correlation between the bio-markers levels and drugs in MDT and propose an optimal drug regimen through these optimal control studies. We use the Newton's Gradient Method for the optimal control studies.

Abstract: We investigate an expected utility maximization problem under model uncertainty in a one-period financial market. We capture model uncertainty by replacing the baseline model $\mathbb{P}$ with an adverse choice from a Wasserstein ball of radius $k$ around $\mathbb{P}$ in the space of probability measures and consider the corresponding Wasserstein distributionally robust optimization problem. We show that optimal solutions converge to the uniform diversification strategy when uncertainty is increasingly large, i.e. when the radius $k$ tends to infinity.

Abstract: We study the effect of using high-resolution elevation data on the selection of the most fuel-efficient (greenest) path for different trucks in various urban environments. We adapt a variant of the Comprehensive Modal Emission Model (CMEM) to show that the optimal speed and the greenest path are slope dependent (dynamic). When there are no elevation changes in a road network, the most fuel-efficient path is the shortest path with a constant (static) optimal speed throughout. However, if the network is not flat, then the shortest path is not necessarily the greenest path, and the optimal driving speed is dynamic. We prove that the greenest path converges to an asymptotic greenest path as the payload approaches infinity and that this limiting path is attained for a finite load. In a set of extensive numerical experiments, we benchmark the CO2 emissions reduction of our dynamic speed and the greenest path policies against policies that ignore elevation data. We use the geo-spatial data of 25 major cities across 6 continents, such as Los Angeles, Mexico City, Johannesburg, Athens, Ankara, and Canberra. Our results show that, on average, traversing the greenest path with a dynamic optimal speed policy can reduce the CO2 emissions by 1.19% to 10.15% depending on the city and truck type for a moderate payload. They also demonstrate that the average CO2 reduction of the optimal dynamic speed policy is between 2% to 4% for most of the cities, regardless of the truck type. We confirm that disregarding elevation data yields sub-optimal paths that are significantly less CO2 efficient than the greenest paths.